(Un)boundedness of directional maximal operators through a notion of "Perron capacity'' and an application

TL;DR

This paper introduces the Perron capacity to analyze the boundedness of directional maximal operators, showing that certain sets of slopes lead to unbounded operators on L^p spaces, and resolving a question about lacunarity.

Contribution

It defines Perron capacity and demonstrates its role in determining the unboundedness of directional maximal operators, providing new insights into geometric measure theory.

Findings

Finite Perron capacity implies unboundedness of the maximal operator.

The set _{e} is not finitely lacunary.

Answers a question by A. Stokolos regarding lacunarity.

Abstract

We introduce the notion of \textit{Perron capacity} of a set of slopes $\Omega \subset \mathbb{R}$. Precisely, we prove that if the Perron capacity of $\Omega$ is finite then the directional maximal operator associated $M_\Omega$ is not bounded on $L^p(\mathbb{R}^2)$ for any $1 < p < \infty$. This allows us to prove that the set $$\Omega_{ \boldsymbol{e}} =\left\{ \frac{\cos n}{n}: n\in \mathbb{N}^* \right\}$$ is not finitely lacunary which answers a question raised by A. Stokolos.